Can quantum measurements be the origin of thermodynamic arrow of time? We can practically consider that the microscopic interactions are symmetric with respect to time(as we can neglect weak force for many cases which is the only interaction that can violate $T$ symmetry). So I thought that the asymmetry might be due to the irreversibility of quantum measurements. But this is only applicable for interpretations where wave function collapses like Copenhagen etc. What is the answer to this question in Many-worlds interpretation, Consistent histories, etc? Also in this page, they gave that the initial conditions of the universe are the reason for $T$ asymmetry in the 2nd law of thermodynamics. But I am not sure what they mean. Do they mean that the universe had a very low entropy at the beginning?
 A: 
Do they mean that the universe had a very low entropy at the beginning?

Yes, the universe had a very low entropy immediately after the Big Bang. It was filled with a very uniform distribution of very energetic (very "hot") fundamental particles. Due to the effect of gravity, this uniform distribution is actually a highly improbable state, and so has an extremely low entropy. So the universe started in a low entropy state, and its entropy has been increasing ever since, following the second law of thermodynamics.
The natural follow-up question, which is very interesting, is why did the early universe have such a low entropy to start with ? Was this inevitable, or is it an unusual and unlikely feature of our particular universe ? Since we have no other universes to compare ours with, this is a very difficult question to answer !
A: No, because "measurements" and "wave function collapse" are not part of the system modeled by QM; they're interpretations. As such, no physical phenomena can be effects of them.
A: Entropy is Macroscopic
As RogerJBarlow mentions in a comment, there is no need to invoke QM to explain the "arrow of time".  To use the exact analogy, consider an "ideal billiard table" with the standard collection of 15 balls.  Now, the macro state in which all balls are arranged in the starting triangle is very improbable, because there are only a few "micro-states" (permutations and rotations of the balls) which correspond to this macro state.  Further, with typical energy inputs to the table, there are many ways to move from this macro state to other macro states, but not very many ways to move from other macro states to this one.  That makes it a "high entropy" state.
On the other hand, if we consider the macro state "every pocket has at least 2 balls closer to it than any other pocket", we see that there are very many micro states in this macro state.  There are also many transitions which lead into this state, and the sheer number of micro states means that many transitions will cause you to never leave this macro state.  Thus, this is a "low entropy" state, and one of the more likely outcomes in a typical game of pool.
There is no entanglement, superposition, tunnelling, wave function collapse or any other QM phenomenon required to analyze the entropic behavior of this system, or to guess between a sequence of snapshots which direction indicates the arrow of time.  Replace the billiard balls with gas molecules, and you start to look much like thermodynamics.
